On Sun, 17 Nov 2002 19:46:09 +0100, Glen M. Sizemore wrote:
> 1.) a p value is given by (pA/B), where a is data= or more extreme, and
> b is the null hypothesis. The "/" is, of course, the conditional sign.
Then we agree on this.
> 2.) to say that a p value represents the probability that you will get
> those data by chance is to say that it quantitatively reflects the
> probability that the null hypothesis is false, which is p(A/B)=p(B/A).
The original argument for the p-value was "modus tollens":
(H0 predicts outcome) & (outcome is not observed) => (H0 must be false)
This is of course a valid statement. What it amount to saying is that
we can reject H0 on the grounds that we did not get the predicted result.
And we can compute what results H0 predicts.
So the case for H0 is not based on inverting conditionality. Actually,
the proponents of the p-value would claim that asserting a probability
to a proposition like H0 is meaningless, as only stochasic variables
can have a pdf. So p(H0 | data) or p(H0 | data or more extreme data) has
no meaning as H0 cannot have a probability associated to it. H0 is
a proposition that are either true or false.
A better picture is the mass of Saturn. To a Bayesian, our knowledge
about this mass can be described by a pdf. To a non-bayesian, the mass
of Saturn is a natural constant. What is random is our measurements
of this mass, but only random variables can have a pdf. Saturn only has
one mass, but each time we measure the mass we get slightly different
results. This variability can be described by a pdf.
So the case for H0 is spesifically based on rejecting hypothesis by Modus
Tollens (cf. Karl Popper). There is no implication of "inverted
conditionality" in the argument, as it would be meaningless by definition.
The argument is modus tolles, which is valid.
Note that prob(effect size | H0) actually describes how H0 predicted the
effect size. But since it evaluates to zero for continous data, we must
integrate over a certain range of effect sizes. What range is natural to
chose? Fisher claimed that the natural range is from the effect size that
was observed to infinity. This is the p-value. This made Harold Jeffreys
state that "what the use of H0 implies then, is that a hypothesis that may
or may not be true is rejected because it did not predict something
that did not happen."
The difference between Bayesian statistics and classical statistics
is thus philosophically deeper than that of conditional probability.
Classical statistics is based on falsification by deduction. Bayesian
statistics is based on Occham's razor by induction. Classical statistics
only assigns probabilities to random variables. Bayesian statistics
assigns probabilities to any proposition.
Sturla Molden